CalcRef

Limits & Continuity · Limit Laws

Limit of a Power

lim⁡x→a[f(x)]n=[lim⁡x→af(x)]n\lim_{x \to a} [f(x)]^n = \left[\lim_{x \to a} f(x)\right]^n

The limit of a power equals the power of the limit, for any positive integer n.

Conditions. lim f(x) must exist and n is a positive integer.

Worked examples

Find lim(x→2) (x+1)³.
  1. [lim(x→2)(x+1)]³ = 3³

Answer: 27

Find lim(x→-1) (x²+2)².
  1. [lim(x→-1)(x²+2)]² = (1+2)² = 3²

Answer: 9

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